Topological and Metric Properties

[Bachelier]

Can someone explain the difference between the two?

2 topo spaces are isometric if they have the same metric properties and homeometric if they have the same topological properties.

If a space is homeo it is iso, but not vice verse. Which begs the conclusion that every topological property is a metric one, but not every metric is topological.
Is it that a property is metric if it is related to the metric used on the space. That's how in the same space like R, we can prove that cauchiness is not topological by changing the metric. So is Cauchiness a metric property? What about Boundedness?
However continuity of a function is topological. Meaning it is mainly linked to the space we're working in?

I'd appreciate some input and especially examples of different metric and topological properties.

 

[HallsofIvy]

"metric" spaces form a subset of all topological spaces- that is a metric space is a topological space in which the topology is defined by a specfic metric. But not all topological spaces are metric spaces. It is NOT true that "if a space is homeo then it is iso". In fact, it doesn't even make sense to talk about a space being "homeo" or "iso". Two spaces may be homeomorphic or isomorphic to one another.

As for "every topological property is a metric one but not every metric (property) is topological". Yes, a metric space is a topological space so every property defined for all topological spaces applies to metric spaces also. No, not every topological space admits a metric so some properties of metric spaces (such as "bounded") applies to general topological spaces. If, by "Cauchiness" (a word I hope I never use again!) you mean "the Cauchy Criterion", that is, necessarily a metric property since you require that $|a_n- a_m|$ go to 0 which requires a metric.

Note that "compactness" is a topological property but "boundedness" is not so it only makes sense to ask if all compact sets are bounded in metric spaces.

 

[lavinia]

Can someone explain the difference between the two?

2 topo spaces are isometric if they have the same metric properties and homeometric if they have the same topological properties.

If a space is homeo it is iso, but not vice verse. Which begs the conclusion that every topological property is a metric one, but not every metric is topological.
Is it that a property is metric if it is related to the metric used on the space. That's how in the same space like R, we can prove that cauchiness is not topological by changing the metric. So is Cauchiness a metric property? What about Boundedness?
However continuity of a function is topological. Meaning it is mainly linked to the space we're working in?

I'd appreciate some input and especially examples of different metric and topological properties.

Two spaces can be homeo without being iso.

 

[Bachelier]

I get it. Because if 2 homeo TPs have a different metric, then they won't be iso.

Thanks

 

[blue_raver22]

The difference between topological and metric properties lies in the way they describe the properties of a space. A topological property is a characteristic of a space that remains unchanged under homeomorphisms, which are continuous and bijective functions between spaces. In other words, if two spaces are homeomorphic, they have the same topological properties. These properties are related to the geometric structure of the space, such as connectedness, compactness, and separation.

On the other hand, a metric property is a characteristic of a space that is determined by the metric used on that space. A metric is a function that assigns a distance between any two points in a space. Therefore, a metric property is related to the distance between points in a space, such as continuity, convergence, and completeness.

In the example given, two topological spaces are isometric if they have the same metric properties, meaning that the distance between points in the two spaces is preserved under the isometry. On the other hand, two topological spaces are homeometric if they have the same topological properties, meaning that the geometric structure of the two spaces is preserved under the homeomorphism.

It is important to note that not all topological properties are metric properties. For example, compactness is a topological property that is not determined by the metric. In other words, a compact space may have different metrics that induce different topologies.

In regards to your question about Cauchy-ness and boundedness, these are both metric properties. Cauchy-ness refers to the convergence of a sequence, which is determined by the metric used on the space. Boundedness, on the other hand, refers to the size of the space and is also determined by the metric.

Continuity of a function is a topological property because it is related to the structure of the space and not just the metric used on the space.

Some examples of different metric and topological properties include completeness, connectedness, compactness, and separability. Completeness is a metric property that refers to the existence of limits for all Cauchy sequences in a space. Connectedness and compactness are topological properties that describe the geometric structure of a space. Separability is a topological property that refers to the existence of a countable dense subset in a space.

In conclusion, topological and metric properties describe different aspects of a space and are determined by different factors. While some properties may